K-Theory as a special $\lambda$-ring I wonder if there is a nice and short proof that the $K$-theory of a topological space is a special $\lambda$-ring without invoking the splitting principle and alike. Is it possible to show directly that $\lambda^k(V \otimes W)$ and $P_k(\lambda^1(V),...,\lambda^k(V),\lambda^1(W),...,\lambda^k(W))$ are stabily equivalent, without making unnatural choices? The same question for $\lambda^i(\lambda^k(V))$ and $P_{ik}(\lambda^1(V),...,\lambda^{ik}(V))$. It would be nice if there is some natural isomorphism on the level of vector spaces, which then may be glued to an isomorphism between vector bundles, perhaps with some extra summands on both sides which vanish in K-theory. Note that an affirmative answer would, in particular, answer this question by Darij Grinberg.
I hope that my question is clear enough although it is not precise. On the other hand, there is a precise generalization: Let $A$ be a topological ring, perhaps a Banach algebra. Is the Grothendieck ring of topological $A$-module bundles over a fixed $X$ a special $\lambda$-ring?
 A: All vector bundles are pull backs from the appropriate infinite Grassmanians/ classifying spaces. So it is sufficient to answer the question there.
But the category of vector bundles on such a space is just the category of representations of the appropriate group. So we just need to prove these for the special cases where the category is the category of representations of $GL_a \times GL_b$, and $V$ and $W$ are the standard representations of $GL_a$ and $GL_b$ respectively.
But the representation ring of those groups are well-known to be polynomial algebras generated by the $\lambda^i$s! So there is some polynomial formula. What polynomials must these be? By pulling back the representations to the representation ring of the maximal torus, we see that these polynomials must satisfy the identities that are usually taken to define them.
We can see immediately that this proof gives a natural isomorphism on the level of vector spaces, in that it gives a natural transformation between two endofunctors of the category of vector spaces where maps are isomorphisms, alias, an isomorphism between representations of $GL_n$.
